Bài 28 trang 205 SGK Giải tích 12 Nâng cao
Bài 28 trang 205 SGK Giải tích 12 Nâng cao Viết các số phức sau dưới dạng lượng giác ...
Bài 28 trang 205 SGK Giải tích 12 Nâng cao
Viết các số phức sau dưới dạng lượng giác
Bài 28. Viết các số phức sau dưới dạng lượng giác:
(eqalign{
& a),,1 - isqrt 3 ;,,1 + i;,,(1 - isqrt 3 )(1 + i);,,{{1 - isqrt 3 } over {1 + i}}; cr
& b),,2ileft( {sqrt 3 - i}
ight); cr
& c),,{1 over {2 + 2i}}; cr
& d),,z = sin varphi + icos varphi ,(varphi inmathbb R) cr} )
Giải
(eqalign{
& a),,1 - isqrt 3 = 2left( {{1 over 2} - {{sqrt 3 } over 2}i}
ight) = 2left( {cos left( { - {pi over 3}}
ight) + isin left( { - {pi over 3}}
ight)}
ight);,,,,, cr
& ,,,,,,,,1 + i = sqrt 2 left( {{1 over {sqrt 2 }} + {1 over {sqrt 2 }}i}
ight) = sqrt 2 left( {cos left( {{pi over 4}}
ight) + isin left( {{pi over 4}}
ight)}
ight);, cr
& ,,,,,,,,(1 - isqrt 3 )(1 + i) = 2sqrt 2 left( {{1 over 2} - {{sqrt 3 } over 2}i}
ight)left( {{1 over {sqrt 2 }} + {1 over {sqrt 2 }}i}
ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 2sqrt 2 left( {cos left( { - {pi over 3}}
ight) + isin left( { - {pi over 3}}
ight)}
ight)left( {cos {pi over 4} + isin {pi over 4}}
ight) cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 2sqrt 2 left[ {cos left( {{pi over 4} - {pi over 3}}
ight) + isin left( {{pi over 4} - {pi over 3}}
ight)}
ight] cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, = 2sqrt 2 left[ {cos left( { - {pi over {12}}}
ight) + isin left( { - {pi over {12}}}
ight)}
ight];,, cr
& {{1 - isqrt 3 } over {1 + i}} = sqrt 2 left[ {cos left( { - {pi over 3} - {pi over 4}}
ight) + isin left( { - {pi over 3} - {pi over 4}}
ight)}
ight] cr
& ,,,,,,,,,,,,,,,,,,,;;;, = sqrt 2 left[ {cos left( { - {7 over {12}}pi }
ight) + isin left( { - {7 over {12}}pi }
ight)}
ight]; cr
& b),,2i = 2left( {cos {pi over 2} + isin {pi over 2}}
ight) cr
& ,,,,,,,left( {sqrt 3 - i}
ight) = 2left( {{{sqrt 3 } over 2} - {1 over 2}i}
ight) = 2left[ {cos left( { - {pi over 6}}
ight) + isin left( { - {pi over 6}}
ight)}
ight]; cr
& ,,,,,,,2ileft( {sqrt 3 - i}
ight) = 4left[ {cos left( {{pi over 2} - {pi over 6}}
ight) + isin left( {{pi over 2} - {pi over 6}}
ight)}
ight] cr
& ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ;;,= 4left[ {cos left( {{pi over 3}}
ight) + isin left( {{pi over 3}}
ight)}
ight] cr
& c),,2 + 2i = 2sqrt 2 left( {{1 over {sqrt 2 }} + {1 over {sqrt 2 }}i}
ight) = 2sqrt 2 left( {cos {pi over 4} + isin {pi over 4}}
ight), cr
& Rightarrow {1 over {2 + 2i}} = {1 over {2sqrt 2 }}left[ {cos left( { - {pi over 4}}
ight) + isin left( { - {pi over 4}}
ight)}
ight] cr
& d),z = ,sin varphi + icos varphi = ,cos left( {{pi over 2} - varphi }
ight) + isinleft( {{pi over 2} - varphi }
ight)(varphi in mathbb R) cr} )
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